Why are the balls disjoint from each other? My attempt: If there exists an element $g$ in $B(e,\sqrt{2})\bigcap B(f,\sqrt{2})$ then by the triangle inequality, $\sqrt{2}=\left\|e-f\right\|\leq \left\|e-g\right\|+\left\|g-f\right\|=\sqrt{2}+\sqrt{2}=2\sqrt{2}$, but I cannot obtain the contradiction. What would be the appropriate argument to find such contradiction? Thank you.
The $\sqrt 2$ is a typo - you can easily see that they aren't disjoint in the 2-dimensional $x$-$y$ plane if you take $e_1=(1,0)$ and $e_2=(0,1)$.
Just make it smaller - pick a value that will make your triangle inequality argument work.